# Prove that for an ideal-gas reaction, #(dlnK_C^@)/(dT) = (DeltaU^@)/(RT^2)#?

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I found this question particularly difficult, but I figured it out and wanted to share.

I found this question particularly difficult, but I figured it out and wanted to share.

##### 1 Answer

For an ideal gas reaction, we begin with the definition of

#K_C^@ = K_P^@((P^@)/(RTc^@))^(Deltan)#

and differentiate

#(dlnK_C^@)/(dT)#

#= d/(dT)[ln{e^(-(DeltaG^@)/(RT))((P^@)/(RTc^@))^(Deltan)}]#

#= d/(dT)[-(DeltaG^@)/(RT) + Deltanln((P^@)/(RTc^@))]#

#= d/(dT)[-(DeltaG^@)/(RT)] + Deltan(cancel(RTc^@)/cancel(P^@))*-cancel(P^@)/(cancel(Rc^@)T^cancel(2))#

#= stackrel("Product Rule")overbrace((DeltaG^@)/(RT^2) - 1/(RT)(dDeltaG^@)/(dT)) - (Deltan)/T#

By definition, since

#dG^@ = -S^@dT + VdP^@#

or

#dDeltaG^@ = -DeltaS^@dT + DeltaVdP^@#

and acquire

#((delDeltaG^@)/(delT))_(P^@) = (dDeltaG^@)/(dT) = -DeltaS^@#

Then we proceed to acquire the result by noting that

#=> (DeltaG^@)/(RT^2) + (DeltaS^@)/(RT) - (Deltan)/T = (DeltaG^@)/(RT^2) + (TDeltaS^@)/(RT^2) - (DeltanRT)/(RT^2)#

#= (DeltaG^@ + TDeltaS^@ - DeltanRT)/(RT^2) = (DeltaH^@ - cancel(TDeltaS^@ + TDeltaS^@) - DeltanRT)/(RT^2)#

#= (DeltaU^@ + Delta(PV) - DeltanRT)/(RT^2) = (DeltaU^@ + cancel(DeltanRT) - cancel(DeltanRT))/(RT^2)#

#= color(blue)((DeltaU^@)/(RT^2))#